T
E
L
KO
M
NIK
A
, V
ol
.
17
,
No.
5,
O
c
tob
er
20
1
9,
p
p.2
32
7
~
23
34
IS
S
N: 1
69
3
-
6
93
0
,
accr
ed
ited
F
irst
Gr
ad
e b
y K
em
en
r
istekdikti,
Decr
ee
No: 2
1/E/
K
P
T
/20
18
DOI:
10.12928/TE
LK
OM
N
IK
A
.v
1
7
i
5
.
10525
◼
23
27
Rec
ei
v
ed
J
ul
y
8
,
20
1
8
;
Rev
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s
ed
F
e
bruar
y
9
,
20
1
9
;
A
c
c
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arc
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20
1
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Blind
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r
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s
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IF
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n
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p
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s
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d
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b
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b
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Key
w
ords
:
b
l
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d
m
u
l
ti
-
s
i
g
n
a
t
u
re
,
b
l
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d
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a
tu
re
,
d
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,
d
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s
c
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l
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r
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th
m
p
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b
l
e
m
,
i
n
te
g
e
r
fa
c
to
r
i
z
a
ti
o
n
p
ro
b
l
e
m
Copy
righ
t
©
2
0
1
9
Uni
v
e
rsi
t
a
s
Ahm
a
d
D
a
hl
a
n.
All
rig
ht
s
r
e
s
e
rve
d
.
1.
Int
r
o
d
u
ctio
n
O
ne
of
the
i
m
po
r
tan
t
o
bj
ec
ti
v
es
of
the
i
nf
orm
ati
on
s
ec
urit
y
s
y
s
tem
s
i
s
prov
i
di
n
g
au
th
en
t
i
c
ati
on
of
the
el
ec
tr
on
i
c
do
c
um
en
ts
an
d
m
es
s
ag
es
.
Us
ua
l
l
y
d
i
gi
tal
s
i
gn
a
ture
s
c
he
m
es
are
c
on
s
i
de
r
e
d
t
he
m
os
t
i
m
po
r
tan
t
s
ol
ut
i
o
ns
to
m
ee
ti
ng
th
es
e
r
e
qu
i
r
em
en
ts
[1
].
T
he
r
e
wer
e
m
an
y
pro
po
s
a
l
s
f
or
s
i
gn
at
ure
s
c
he
m
es
pu
bl
i
s
h
ed
b
as
ed
o
n
a
s
i
ng
l
e
h
ard
pr
ob
l
em
s
uc
h
as
f
ac
tori
ng
(
F
A
C)
,
di
s
c
r
ete
l
o
ga
r
i
thm
(
DL)
or
e
l
l
i
pt
i
c
c
urv
e
d
i
s
c
r
ete
l
og
ar
i
thm
(
E
CDL
)
probl
em
s
[1]
.
A
l
s
o
d
i
gi
ta
l
s
i
gn
a
ture
s
c
he
m
e
s
f
r
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m
tw
o
d
i
f
f
i
c
ul
t p
r
o
bl
em
s
were
p
r
op
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ed
bu
t
m
os
t o
f
th
em
ha
v
e
prov
e
d t
o
be
no
t
as
s
ec
ure
as
c
l
ai
m
ed
[2
-
4].
In
v
ario
us
t
y
pe
s
of
el
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tr
o
ni
c
tr
an
s
ac
t
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on
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tem
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s
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t
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nt
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at
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are
al
w
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s
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m
the
b
l
i
nd
s
i
gn
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r
e
s
c
he
m
es
are
us
ed
[5
-
7
].
T
he
prope
r
ti
es
of
the
bl
i
nd
s
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g
na
tures
are
th
e
s
i
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r
c
an
no
t
to
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e
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th
e
do
c
u
m
en
t
du
r
i
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proc
es
s
of
s
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at
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ge
ne
r
at
i
on
and
t
h
e
s
i
gn
er
c
an
no
t
c
orr
el
ate
the
s
i
gn
ed
d
oc
um
en
t
wi
th
the
ac
t o
f
s
i
gn
i
ng
.
T
he
r
e
w
ere
m
an
y
pr
op
os
a
l
s
f
or
bl
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nd
s
i
gn
at
ure
s
c
h
e
m
e
s
pu
b
l
i
s
h
ed
ba
s
e
d
on
a
s
i
n
gl
e
ha
r
d
pro
bl
em
s
uc
h
as
F
A
C,
DL
or
E
CD
L
pro
bl
em
s
[
8
-
12
].
A
l
l
of
the
m
r
em
ai
n
s
ec
ure
an
d
are
r
es
i
s
tan
t
to
att
ac
k
s
.
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w
e
v
er,
i
f
on
e
f
i
n
ds
a
s
ol
u
ti
o
n
f
or
the
un
de
r
l
y
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ng
ha
r
d
p
r
ob
l
em
he
nc
e
break
the
c
orr
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p
on
d
i
n
g
s
i
gn
at
ure
s
c
he
m
es
ea
s
i
l
y
.
In
[
6,
7
]
pro
po
s
ed
b
l
i
nd
s
i
gn
a
ture
s
c
he
m
es
,
whi
c
h
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e
qu
i
r
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i
m
ul
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us
s
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v
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ng
of
t
w
o
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en
de
n
t
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f
f
i
c
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t
prob
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e
m
s
.
How
e
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the
y
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v
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hi
gh
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i
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B
l
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m
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na
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s
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atu
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es
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n
whi
c
h
th
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grou
p
of
s
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ers
(
B
)
do
no
t
k
no
w
w
h
at
the
y
are
s
i
gn
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ng
,
th
us
th
e
term
“
bl
i
nd
”
.
S
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h
s
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g
na
t
ures
are
po
s
s
i
b
l
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be
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a
us
e
t
he
c
o
nte
n
t
of
the
m
es
s
ag
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M
h
as
be
e
n
“
bl
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Mʹ
be
f
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t
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m
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i
s
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de
d
to
the
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ti
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s
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gn
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T
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s
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th
e
s
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gn
i
ng
c
o
l
l
ec
t
i
v
e
s
i
gn
ed
Mʹ
an
d
n
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M
.
S
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c
i
f
i
c
a
l
l
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,
the
us
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A
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ed
s
the
c
ol
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v
e
B
to
s
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M
;
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o
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A
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h
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t
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s
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to
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he
n
pr
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de
s
t
he
bl
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Mʹ
to
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to
s
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g
n.
A
f
ter
r
ec
ei
v
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t
h
e
s
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ed
Mʹ
,
A
un
b
l
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he
m
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s
ag
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to
o
bta
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n
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s
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t
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f
or
M
.
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he
r
ef
ore,
A
ha
s
a
s
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e
f
or
M
w
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t
ho
u
t
prov
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d
i
ng
B
wi
th
i
nf
orm
ati
on
on
M
.
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
5,
O
c
tob
er 20
19
:
23
2
7
-
23
34
2328
In
1
99
9,
P
o
pe
s
c
u
[
1
3
]
pr
es
en
te
d
bl
i
nd
m
ul
ti
-
s
i
gn
at
ures
ba
s
e
d
on
e
l
l
i
pt
i
c
c
ur
v
es
.
In
20
05
,
Cho
w
et
a
l
.
pro
p
os
ed
t
w
o
b
l
i
n
d
s
i
gn
atu
r
e
s
c
he
m
es
pa
r
ti
al
l
y
ba
s
e
d
on
B
i
l
i
ne
ar
P
ai
r
i
ng
s
[
14
].
In
20
11
,
Mo
l
do
v
y
a
n
[
15
]
pres
en
te
d
a
b
l
i
nd
s
i
gn
atu
r
e
s
c
h
em
e
ba
s
ed
on
the
G
O
S
T
R34.
1
0
-
20
0
1
s
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g
na
ture
s
tan
da
r
d.
In
20
12
,
N
gu
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e
n
an
d
Da
ng
[
16
]
pr
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d
en
ha
nc
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d
s
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f
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v
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ti
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I
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V
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on
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g
er
F
ac
t
ori
z
ati
on
an
d
Di
s
c
r
ete
-
Lo
g
arit
hm
P
r
ob
l
e
m
s
[1
7
].
In
20
13
,
P
a
nd
a
et
al
.
r
es
ea
r
c
h
ed
b
l
i
nd
s
i
gn
i
ng
au
th
ori
z
at
i
on
s
i
n
el
ec
tr
on
i
c
v
oti
ng
proc
es
s
es
[
18
].
I
n
20
1
4,
Hua
S
un
et
a
l
.
prop
os
ed
Ne
w
C
erti
f
i
c
ate
l
es
s
B
l
i
n
d
Ri
ng
S
i
gn
atu
r
e
S
c
he
m
e
[
19
].
In
20
1
6,
S
hi
l
b
a
y
e
h
e
t
al
.
propos
e
d
s
ec
urit
y
s
c
he
m
es
f
or
el
ec
tr
on
i
c
v
ot
i
ng
pr
oc
es
s
es
[2
0
].
In
2
01
7,
M
i
n
h
et
al
.
propos
e
d
Ne
w
B
l
i
n
d
S
i
gn
atu
r
e
P
r
oto
c
ol
s
B
as
ed
o
n
a
Ne
w
Har
d
P
r
ob
l
em
[21
];
S
al
om
e
J
a
m
es
et
al
.
pro
po
s
ed
Id
en
t
i
t
y
-
B
as
ed
B
l
i
n
d
S
i
g
na
ture
S
c
he
m
e
wi
th
Me
s
s
ag
e Re
c
o
v
er
y
[2
2
].
In
thi
s
p
ap
er,
we
prop
os
e
a
ne
w
di
g
i
ta
l
s
i
g
na
t
ure
s
c
he
m
e
fr
o
m
tw
o
di
f
f
i
c
ul
t
pr
ob
l
em
s
ba
s
ed
on
th
e
RS
A
di
g
i
t
al
s
i
gn
at
ure
s
c
he
m
e
[23
]
an
d
t
he
S
c
hn
orr
d
i
gi
tal
s
i
gn
at
ure
s
c
he
m
e
[24]
.
W
e
ex
pa
nd
ou
r
f
un
c
ti
on
al
i
t
y
to
c
on
s
tr
uc
t
t
he
bl
i
nd
s
i
g
na
tur
e
s
c
he
m
e
a
nd
t
he
bl
i
n
d
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e
.
T
hi
s
he
l
ps
ne
w
bl
i
nd
d
i
gi
tal
s
i
g
na
tur
e
s
c
he
m
es
i
nh
erit
s
om
e
adv
an
t
ag
es
of
the
s
ec
urit
y
of
t
he
s
i
gn
atu
r
e
s
c
he
m
e
s
tha
t
ha
d
b
ee
n
pro
v
en
i
n
prac
ti
c
e
.
T
he
organ
i
z
ati
on
of
the
pa
pe
r
i
s
as
f
ol
l
o
w
s
:
s
ec
ti
on
2
pro
v
i
de
s
t
h
e
r
el
ate
d
the
orie
s
an
d
s
c
he
m
es
.
In
s
ec
ti
on
3
,
w
e
s
ha
l
l
de
s
i
g
n
a
ne
w
s
i
g
na
tur
e
s
c
he
m
e,
whi
c
h
r
e
qu
i
r
es
the
s
i
m
ul
tan
eo
us
break
i
ng
of
the
f
ac
tori
z
ati
on
an
d
di
s
c
r
ete
l
og
arit
hm
.
W
e
ex
pa
n
d
ou
r
f
un
c
ti
on
a
l
i
t
y
t
o
c
on
s
tr
uc
t
a
ne
w
b
l
i
nd
s
i
gn
at
ure
s
c
he
m
e
an
d
a
ne
w
b
l
i
n
d
m
ul
ti
-
s
i
g
na
ture
s
c
he
m
e.
In
th
e l
as
t s
ec
ti
o
n,
th
e c
on
c
l
us
i
o
n o
f
ou
r
r
es
e
arc
h
w
ork
wi
l
l
b
e p
r
es
e
nte
d.
2.
Rel
ated
T
h
eorie
s
and
S
chemes
T
he
f
ol
l
o
w
i
ng
no
ta
ti
o
ns
are
us
ed
:
-
p
i
s
a
prim
e
nu
m
be
r
,
w
i
t
h
s
tr
uc
ture
p
=2
n
+
1,
n
q
q
=
w
i
th
,
qq
are
the
s
tr
on
g
prim
e
nu
m
be
r
s
[2
5]
-
H
i
s
a
c
o
l
l
i
s
i
o
n
-
r
es
i
s
tan
t
ha
s
h f
un
c
ti
on
-
i
s
a
g
en
era
tor of
order
n
o
v
er
*
p
Z
-
q
|
p
−
1
:
q
i
s
th
e
di
v
i
s
or of
p
-
1
-
()
n
i
s
th
e
E
u
l
er f
un
c
ti
on
-
e
i
s
th
e p
ub
l
i
c
k
e
y
an
d d
i
s
the
s
ec
r
et
k
e
y
i
n R
S
A
2
.1.
D
isc
r
ete L
o
g
a
r
it
h
m P
r
o
b
lem (
DL
P
)
[
2
6]
T
hi
s
probl
em
i
s
de
s
c
r
i
b
ed
a
s
f
ol
l
o
w
s
:
G
i
v
e
n
an
i
ns
t
an
c
e
(
,
,
,
)
y
p
q
,
w
h
ere
m
o
d
x
yp
=
f
or s
o
m
e
*
,
q
xZ
to
de
r
i
v
e
x
.
2
.
2
.
Int
eg
er
Fa
cto
r
iz
atio
n
P
r
o
b
lem
[
26
]
T
he
i
nte
ge
r
f
ac
tori
z
ati
on
p
r
ob
l
em
i
s
the
f
ol
l
o
w
i
ng
:
g
i
v
en
a
p
os
i
t
i
v
e
i
nt
eg
er
n
,
f
i
nd
i
ts
prim
e
f
ac
tori
z
a
ti
on
,
i
.e.
,
f
i
nd
p
ai
r
wi
s
e
d
i
s
ti
nc
t
prim
es
i
p
an
d
p
os
i
t
i
v
e
i
nte
ge
r
s
i
e
s
uc
h
tha
t
12
12
.
.
.
.
k
e
ee
k
n
p
p
p
=
T
he
eth
r
oo
t
prob
l
em
i
s
the
f
ol
l
o
w
i
ng
:
G
i
v
en
a
grou
p
G
of
un
k
no
wn
ord
er,
a
po
s
i
t
i
v
e i
nte
g
er
eG
an
d a
n e
l
e
m
en
t
,
aG
f
i
nd
a
n e
l
em
en
t
,
bG
s
uc
h t
ha
t
.
e
ba
=
If
,
n
GZ
=
wi
th
n
be
i
n
g
th
e
produc
t
t
w
o
prim
es
p
a
nd
q
,
an
d
t
he
c
o
nd
i
ti
o
n
t
ha
t
,
bG
i
s
r
ep
l
ac
e
d
b
y
,
n
bZ
w
e
g
et
t
he
R
S
A
pro
bl
em
.
In
th
i
s
c
as
e
th
e
ord
er
of
the
gro
up
c
an
b
e
f
ou
n
d
b
y
f
ac
tori
n
g
n
.
2
.
3
.
RS
A
S
ign
atu
r
e
S
ch
eme
2.3
.1
.
Ke
y
G
ene
r
atio
n
-
Choo
s
e
l
arg
e d
i
s
ti
nc
t
prim
es
p
an
d
q
,
an
d c
om
pu
te
n
=
pq
.
-
Choo
s
e
e
s
uc
h t
ha
t
g
c
d
(
,
(
)
)
1
.
en
=
T
he
p
ai
r
(
,
)
ne
i
s
pu
b
l
i
s
h
ed
as
th
e
pu
b
l
i
c
k
e
y
.
-
Com
pu
te
d
s
uc
h t
ha
t
1
m
o
d
(
)
.
d
e
n
=
T
he
pa
i
r
(
,
)
nd
i
s
us
ed
as
th
e s
ec
r
et
k
e
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
B
l
i
nd
mu
l
ti
-
s
i
gn
a
ture s
c
he
me
b
as
ed
o
n
f
ac
tori
n
g
....
(
Duc
Ngu
y
e
n Tan)
2329
-
Let
H
an
d
c
om
pu
ted
f
r
o
m
the
s
i
gn
e
d
do
c
um
en
t
M
.
It
i
s
prov
en
t
ha
t
ob
t
ai
ni
n
g
a
priv
ate
k
e
y
f
r
o
m
th
e p
ub
l
i
c
k
e
y
i
s
v
er
y
di
f
f
i
c
ul
t u
n
l
es
s
y
o
u k
no
w
th
e f
ac
tori
z
a
ti
o
n o
f
n
[
1,
2
3].
2.3
.2
.
S
ign
atu
r
e
G
en
er
ati
o
n
P
r
o
c
edu
r
e
T
he
s
i
gn
a
ture
pa
i
r
c
a
n
be
c
o
m
pu
ted
ea
s
i
l
y
b
y
a
s
i
g
n
er
w
h
o
k
no
w
s
t
he
m
es
s
ag
e
a
nd
s
ec
r
et
k
e
y
(
p
,
q
,
d
)
as
f
ol
l
o
w
s
:
m
o
d
.
d
S
H
n
=
T
he
n t
he
(
,
)
MS
i
s
th
e d
i
g
i
ta
l
s
i
gn
at
ure of
M
.
2.3
.3
.
S
ign
atu
r
e
V
e
r
if
ica
t
i
o
n
P
r
o
c
edu
r
e
It
i
s
ea
s
y
t
o
v
erif
y
th
at
(
,
)
MS
i
s
v
a
l
i
d
b
y
c
he
c
k
i
ng
i
f
the
f
ol
l
o
wi
n
g
eq
u
al
i
t
y
h
ol
ds
:
m
o
d
,
e
S
H
n
=
where
eq
u
al
i
t
y
f
ol
l
o
w
s
be
c
au
s
e
1
m
o
d
(
)
e
d
n
.
T
he
ha
s
h
f
un
c
ti
on
H
i
s
us
ed
to
en
ha
nc
e s
ec
urit
y
an
d
ef
f
i
c
i
en
c
y
.
2.4
.
S
chn
o
r
r
S
ign
atu
r
e
S
chem
e
2.4
.1
.
Ke
y
G
ene
r
atio
n
-
Choo
s
e r
an
d
om
l
y
a s
ec
r
et
k
e
y
x
wi
th
*
.
q
xZ
Com
pu
te
m
od
.
x
yp
=
-
Let
H
an
d
c
om
pu
ted
f
r
o
m
the
s
i
gn
e
d
do
c
um
en
t
M
.
T
he
pu
bl
i
c
k
e
y
i
s
(
,
,
)
.
py
T
he
s
ec
r
et
k
e
y
i
s
x
.
2.4
.2
.
S
ign
atu
r
e
G
en
er
ati
o
n
P
r
o
c
edu
r
e
T
o s
i
gn
a
m
es
s
ag
e
M
th
e s
i
gn
er p
erf
or
m
s
th
e f
ol
l
o
wi
ng
s
tep
s
:
-
Choo
s
e
a ran
do
m
k
s
uc
h t
ha
t
1
1
.
kq
−
Com
pu
te
m
o
d
.
k
Rp
=
-
Com
pu
te
(
)
.
E
H
M
R
=
Com
pu
te
m
o
d
S
k
x
E
q
=−
.
T
he
n
th
e
pa
i
r
(
,
)
ES
i
s
the
d
i
g
i
tal
s
i
gn
at
ure of
M
. T
he
s
i
gn
er r
ep
ea
ts
th
es
e s
t
ep
s
f
or ev
ery
s
i
g
na
t
ure.
2.4
.3
.
S
ign
atu
r
e
V
e
r
if
ica
t
i
o
n
P
r
o
c
edu
r
e
A
s
i
g
na
tur
e
(
,
)
ES
of
a
m
es
s
ag
e
M
i
s
v
erif
i
e
d a
s
f
ol
l
o
w
s
:
-
Com
pu
te
*
*
*
m
o
d
;
(
||
)
.
SE
R
y
p
E
H
M
R
==
-
Com
pa
r
e
the
v
al
u
es
E
*
a
nd
E
.
If
*
EE
=
th
en
s
i
gn
atu
r
e
i
s
v
a
l
i
d.
T
he
v
er
i
f
i
er
ac
c
ep
ts
a
s
i
gn
at
ure
i
f
al
l
c
o
nd
i
ti
on
s
ar
e
s
ati
s
f
i
ed
an
d
r
ej
ec
ts
i
t o
th
erw
i
s
e.
3.
Bl
ind
S
ign
atu
r
e
S
che
me
b
as
ed o
n
Dif
f
icult
y
of
S
o
lv
ing
S
imu
lt
aneo
u
sl
y
T
w
o
Dif
f
icult
P
r
o
b
le
ms
3.1
. N
ew
S
ign
atu
r
e
S
che
me
b
as
ed o
n
T
w
o
Dif
f
icult
P
r
o
b
lem
s
T
o
de
s
i
gn
the
ne
w
b
l
i
nd
s
i
gn
at
ure
s
c
he
m
e
an
d
bl
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e,
we
f
i
r
s
t
propos
e
a
ne
w
d
i
g
i
ta
l
s
i
gn
atu
r
e
s
c
he
m
e
as
a
ba
s
i
c
s
tr
uc
ture
of
o
ur
d
e
v
el
op
i
ng
bl
i
nd
s
i
g
na
ture
s
c
he
m
es
.
B
r
ea
k
i
ng
the
m
od
i
f
i
ed
s
i
gn
a
ture
s
c
he
m
es
de
s
c
r
i
be
d
be
l
o
w
r
eq
ui
r
e
s
s
i
m
u
l
tan
e
ou
s
s
ol
v
i
ng
t
wo
di
f
f
erent
di
f
f
i
c
ul
t
pro
bl
em
s
,
c
o
m
pu
ti
ng
d
i
s
c
r
ete
l
og
arit
hm
i
n
the
gro
un
d
f
i
el
d
()
G
F
p
and
f
ac
tori
n
g
n
. In
t
hi
s
s
i
gn
atu
r
e
s
c
he
m
e,
p
i
s
a
pri
m
e n
um
be
r
, wi
th
s
tr
uc
ture
21
pn
=+
.
T
he
f
ol
l
o
wi
ng
m
od
i
f
i
c
ati
o
n
s
ha
v
e
be
e
n
propos
e
d
t
o
de
s
i
gn
the
n
e
w
ba
s
i
c
s
i
g
na
ture
s
c
he
m
e
:
i
s
us
e
d
a
v
a
l
u
e
ha
v
i
ng
ord
er
e
qu
a
l
to
n
m
od
u
l
o
p
;
ad
di
t
i
o
na
l
e
l
em
en
t
e
of
the
p
ub
l
i
c
k
e
y
;
ad
d
i
t
i
on
al
e
l
em
en
t
d
of
th
e
pri
v
ate
k
e
y
;
i
n
s
tea
d
of
the
v
a
l
u
e
S
i
n
the
s
i
gn
atu
r
e
v
erif
i
c
at
i
on
eq
ua
t
i
on
i
t
i
s
i
n
tr
od
uc
ed
the
v
al
u
e
.
e
S
T
he
v
a
l
ue
s
e
an
d
d
are
g
en
er
ate
d
l
i
k
e
i
n
the
R
S
A
c
r
y
p
tos
y
s
t
em
.
A
s
the
v
al
ue
e
i
t
i
s
s
el
ec
t
ed
a
s
m
al
l
n
um
be
r
(
ha
v
i
n
g
s
i
z
e
f
r
om
16
to
32
bi
ts
)
tha
t
i
s
r
el
at
i
v
el
y
prim
e
to
(
)
(
1
)
(
1
)
.
n
q
q
=
−
−
T
he
v
al
u
e
d
i
s
c
o
m
pu
ted
as
f
ol
l
o
w
s
1
m
od
(
)
.
d
e
n
−
=
T
he
proc
es
s
of
th
e b
as
i
c
s
t
r
uc
ture i
s
d
es
c
r
i
be
f
ol
l
o
wi
n
g:
3.1
.1
. Ke
y
g
ene
r
atio
n
-
Choo
s
e r
an
d
om
l
y
an
i
nt
eg
e
r
n
eZ
s
uc
h t
ha
t
g
c
d
(
,
)
1
.
en
=
-
Cal
c
u
l
ate
a
s
ec
r
et
d
s
uc
h t
ha
t
1
m
o
d
(
)
e
d
n
.
-
Choo
s
e
r
a
nd
om
l
y
a
s
ec
r
et
k
e
y
x
wi
th
*
.
p
xZ
Com
pu
te
m
od
.
x
yp
=
T
he
pu
b
l
i
c
k
e
y
i
s
(
e,
,
y
)
. T
he
s
ec
r
et
k
e
y
i
s
(
x
, d
).
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
5,
O
c
tob
er 20
19
:
23
2
7
-
23
34
2330
3.1
.2
. S
ign
atu
r
e
G
en
er
ati
o
n
P
r
o
c
edu
r
e
-
Com
pu
te
m
od
,
k
Rp
=
w
h
ere
k
i
s
a
s
ec
r
et
r
an
d
om
nu
m
be
r
,
11
kn
−
-
Com
pu
te
(
||
)
E
H
M
R
=
-
Cal
c
u
l
ate
the
v
al
u
e
S
,
s
uc
h
tha
t
m
o
d
e
S
k
x
E
n
=−
,
i
.e.
(
)
m
od
d
S
k
x
E
n
=−
s
uc
h
t
ha
t
m
o
d
.
e
SE
R
y
p
=
T
he
s
i
gn
at
ure i
s
t
he
pa
i
r
(
,
)
ES
.
3.1
.3
. S
ign
atu
r
e
V
e
r
if
ica
t
i
o
n
P
r
o
c
edu
r
e
-
Com
pu
te
*
*
*
m
o
d
;
(
|
|
)
.
e
SE
R
y
p
E
H
M
R
==
-
Com
pa
r
e
the
v
al
u
es
*
E
and
E
.
If
*
EE
=
,
the
n
s
i
gn
atu
r
e
i
s
v
a
l
i
d
.
O
the
r
wi
s
e,
the
s
i
gn
at
ure
is
r
ej
ec
ted
as
i
n
v
al
i
d.
S
ol
v
i
ng
the
d
i
s
c
r
ete
l
og
arit
hm
probl
em
i
n
()
G
F
p
i
s
no
t
s
uf
f
i
c
i
en
t
f
or
break
i
ng
the
m
od
i
f
i
ed
s
c
he
m
e.
No
w
to
bre
ak
the
s
i
g
n
atu
r
e
s
c
he
m
e
i
t
i
s
r
eq
ui
r
ed
to
k
no
w
the
f
ac
tori
z
ati
on
of
n
.
T
he
s
ol
ut
i
on
of
th
e
di
s
c
r
ete
l
og
arit
hm
probl
em
l
ea
ds
to
t
h
e
c
om
pu
tat
i
on
of
the
s
ec
r
et
k
e
y
x
an
d
to
t
h
e
po
s
s
i
b
i
l
i
t
y
to
c
a
l
c
ul
ate
th
e
v
a
l
u
e
*
(
)
m
o
d
S
k
x
E
n
=−
.
Ho
w
e
v
er,
to
c
al
c
ul
a
te
t
he
s
i
g
na
t
ure
S
i
s
r
eq
ui
r
e
d
to
ex
tr
ac
t
the
eth
r
oo
t
m
od
ul
o
n
f
r
om
the
v
al
ue
*
S
.
T
hi
s
r
eq
ui
r
es
f
ac
tori
ng
the
m
od
ul
us
n
.
3.
2
. N
ew
Blind
S
ign
atu
r
e
S
chem
e
T
he
propos
ed
s
i
gn
atu
r
e
s
c
he
m
e
us
i
ng
t
w
o
di
f
f
i
c
ul
t
pr
ob
l
em
s
c
an
be
us
ed
as
a
ba
s
i
c
al
g
orit
hm
f
or
c
on
s
tr
uc
ti
ng
b
l
i
n
d
s
i
g
na
t
ure
s
c
he
m
e
w
h
i
c
h
i
s
s
i
m
i
l
ar
to
t
he
b
l
i
nd
s
i
g
na
ture
s
c
he
m
e
ba
s
ed
on
t
he
R
S
A
[23
]
a
nd
S
c
h
no
r
r
s
i
g
na
t
ure
s
c
he
m
e
s
[24
].
T
hi
s
ap
pr
oa
c
h
wi
l
l
b
e
us
ed
to
de
v
el
op
t
he
f
ol
l
o
wi
ng
bl
i
n
d
s
i
gn
at
ure
s
c
he
m
e,
whi
c
h
r
e
qu
i
r
es
t
he
s
i
m
ul
tan
eo
us
s
ol
v
i
ng
of
th
es
e
two d
i
f
f
i
c
ul
t
prob
l
em
s
.
T
he
r
e a
r
e s
i
x
r
o
un
ds
i
n t
he
b
l
i
nd
s
i
gn
a
ture s
c
he
m
e.
-
Roun
d 1
(
S
i
g
ne
r
B
)
: S
el
ec
ts
a
r
an
do
m
v
a
l
ue
11
kn
−
an
d c
om
pu
tes
m
o
d
.
k
Rp
=
T
he
n h
e s
en
ds
R
to
th
e
us
er A.
-
Roun
d
2
(
Us
er
A
)
:
S
el
ec
t
s
two
r
an
do
m
v
al
u
es
an
d
an
d
c
om
pu
tes
m
o
d
.
R
R
y
p
=
T
he
n u
s
er
A
c
om
pu
tes
(
||
)
E
H
M
R
=
an
d
.
EE
=
−
T
he
n h
e s
en
ds
E
to
t
he
s
i
g
ne
r
B
.
-
Roun
d
3
(
S
i
gn
er
B
)
:
Co
m
pu
tes
the
v
al
ue
m
o
d
,
D
k
x
E
n
=−
s
uc
h
t
ha
t
m
o
d
.
DE
R
y
p
=
T
he
v
al
ue
of
D
i
s
s
en
t
t
o t
h
e u
s
er.
-
Roun
d
4
(
Us
er
A
)
:
S
e
l
ec
t
s
a
r
a
nd
om
v
a
l
u
e
<
n
(
m
as
k
i
ng
f
ac
tor)
,
c
om
pu
tes
the
v
al
ue
(
)
m
o
d
e
D
D
n
=
+
an
d s
e
nd
s
D
to
the
s
i
gn
er
B
.
-
Roun
d
5
(
S
i
gn
er
B
)
:
Co
m
pu
tes
the
v
al
ue
(
)
(
)
m
o
d
d
e
d
d
d
D
D
D
D
n
=
=
+
=
+
an
d
s
en
t
to
t
he
us
er.
-
Roun
d
6
(
Us
er
A
)
:
Com
pu
tes
the
v
al
ue
s
(
,
)
ES
wi
t
h
EE
=
+
a
nd
/
m
o
d
S
D
n
=
.
T
he
s
i
gn
atu
r
e i
s
t
he
p
ai
r
(
,
)
ES
.
V
erif
i
c
at
i
on
of
B
l
i
nd
s
i
gn
at
u
r
e
s
c
he
m
e:
T
he
v
erif
i
c
ati
on
proc
ed
ure
de
s
c
r
i
be
d
i
n
th
e
bl
i
nd
s
i
gn
at
ure
s
c
he
m
e
i
s
the
s
am
e
as
i
n
the
pre
v
i
ou
s
ba
s
i
c
di
gi
tal
s
i
gn
a
ture
s
c
he
m
e,
i
.e.
,
us
i
ng
the
v
erif
i
c
at
i
on
eq
ua
t
i
on
i
s
*
m
o
d
.
e
SE
R
y
p
=
3.
3
. N
ew
Blind
M
u
lt
i
-
s
ign
atu
r
e
S
chem
e
A
s
s
um
e
tha
t
us
er
A
as
k
s
the
e
nti
r
e
grou
p
B
w
h
o
h
as
the
au
t
ho
r
i
t
y
t
o
i
nc
l
u
de
n
s
i
gn
ers
to
s
i
gn
do
c
um
en
t
M
;
ho
w
e
v
er, th
i
s
us
er
do
es
n
ot
want
thi
s
au
th
ori
z
ed
gro
up
to
k
no
w
t
he
c
on
te
nt
of
M
.
F
i
r
s
t,
t
hi
s
us
er
b
l
i
nd
s
the
d
oc
um
en
t
M
,
w
h
i
c
h
b
ec
om
es
do
c
um
en
t
Mʹ
.
T
he
n,
Mʹ
i
s
s
en
t
to
the
au
t
ho
r
i
z
e
d
s
i
gn
i
n
g
gro
up
.
T
hi
s
group
s
i
gn
s
M
ʹ
a
nd
s
en
ds
i
t
b
ac
k
to
the
r
eq
ue
s
ti
ng
us
er.
T
he
n,
the
us
er
un
b
l
i
nd
s
M
ʹ
to
M
an
d
c
he
c
k
s
the
r
ec
ei
v
ed
s
i
gn
atu
r
e.
If
the
s
i
gn
atu
r
e
i
s
v
al
i
d,
t
he
n
the
us
er
ha
s
a
v
al
i
d
s
i
gn
atu
r
e
on
do
c
um
en
t
M.
A
bl
i
nd
m
ul
ti
s
i
gn
atu
r
e
s
c
he
m
e
ha
s
three
pa
r
ti
c
i
pa
n
ts
:
Us
er
A
,
s
i
g
ne
r
s
B
a
nd
a
tr
us
te
d
th
i
r
d
pa
r
t
y
(
T
T
P
)
.
T
he
i
m
pl
em
en
tat
i
on
proc
es
s
f
or
bl
i
nd
s
i
gn
i
ng
the
m
es
s
ag
e
M
i
nc
l
ud
es
th
r
ee
s
c
he
m
es
:
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
B
l
i
nd
mu
l
ti
-
s
i
gn
a
ture s
c
he
me
b
as
ed
o
n
f
ac
tori
n
g
....
(
Duc
Ngu
y
e
n Tan)
2331
3.3.1
.
Ke
y
G
ene
r
atio
n
-
Choo
s
e
r
a
nd
om
l
y
an
i
nt
eg
er
n
eZ
s
uc
h
tha
t
g
c
d
(
,
)
1
.
en
=
Cal
c
u
l
at
e
a
s
ec
r
et
d
s
uc
h
tha
t
1
m
o
d
(
)
e
d
n
.
-
Choo
s
e
r
an
do
m
l
y
a
s
ec
r
et
k
e
y
i
x
wi
t
h
*
.
ip
xZ
Com
pu
te
m
o
d
i
x
i
yp
=
an
d
s
e
nd
i
t
to
T
T
P
to
c
o
m
pu
te
y
of
s
i
gn
i
n
g
g
r
ou
p:
1
m
o
d
,
1
,
2
,
.
.
.
.
n
i
y
y
p
i
n
==
T
he
pu
b
l
i
c
k
e
y
i
s
(
e,
,
y
)
.
T
he
s
ec
r
et
k
e
y
i
s
(
i
x
, d
).
3.
3
.2
. S
ign
atu
r
e
G
en
er
ati
o
n
P
r
o
c
edu
r
e
-
Roun
d
1
(S
i
gn
er
gr
ou
p
B
)
:
e
ac
h
us
er
i
n
t
he
s
i
g
ni
n
g
gr
ou
p
s
el
ec
ts
a
r
an
do
m
v
al
ue
11
i
kn
−
an
d
c
om
pu
tes
m
o
d
.
i
k
i
Rp
=
T
he
n
he
s
e
nd
s
i
R
to
T
T
P
to
c
o
m
pu
te
R
s
uc
h
as
:
1
m
od
1
m
od
m
od
.
n
i
i
n
kp
i
R
R
p
p
=
==
-
Roun
d
2
(
Us
er
A
)
:
S
e
l
ec
ts
t
w
o
r
an
do
m
v
al
ue
s
a
n
d
an
d
c
om
pu
tes
m
o
d
.
R
R
y
p
=
T
he
n c
om
pu
tes
(
||
)
E
H
M
R
=
and
EE
=
−
and
s
en
ds
to
B
.
-
Roun
d
3
(
S
i
gn
er
group
B
)
:
e
ac
h
us
er
i
n
the
s
i
gn
i
ng
grou
p
c
om
pu
te
s
the
v
a
l
ue
m
o
d
,
i
i
i
D
k
x
E
n
=−
s
uc
h
th
at
m
o
d
.
i
D
E
ii
R
y
p
=
T
he
v
a
l
ue
of
i
D
i
s
s
en
t
to
T
T
P
to
c
om
pu
tes
D
s
uc
h a
s
:
1
m
o
d
n
i
i
D
D
n
=
=
an
d s
en
ds
to
A
.
-
Roun
d
4
(
Us
er
A
)
:
S
e
l
ec
ts
a
r
an
d
om
v
al
ue
,
n
c
om
pu
tes
the
v
al
u
e
(
)
m
o
d
e
D
D
n
=
+
an
d s
e
nd
s
to
B
.
-
Roun
d
5
(
S
i
g
ne
r
gr
ou
p
B
)
:
Com
pu
tes
(
)
(
)
m
o
d
d
e
d
d
d
D
D
D
D
n
=
=
+
=
+
an
d
s
en
ds
to
A
.
-
Roun
d
6 (Us
er
A
)
: Com
pu
t
es
th
e
v
a
l
ue
s
(
,
)
ES
wi
th
EE
=
−
and
/
m
o
d
S
D
n
=
-
T
he
s
i
gn
atu
r
e i
s
t
he
p
ai
r
(
,
)
ES
.
3.3
.3
.
S
ign
atu
r
e
V
e
r
if
ica
t
i
o
n
P
r
o
c
edu
r
e
T
he
v
erif
i
c
ati
o
n
proc
e
du
r
e
de
s
c
r
i
be
d
i
n
t
he
bl
i
nd
s
i
gn
atu
r
e
s
c
he
m
e
i
s
the
s
am
e
as
i
n
the
pre
v
i
ou
s
ba
s
i
c
di
g
i
ta
l
s
i
gn
atu
r
e
s
c
he
m
e,
i
.e.
,
us
i
n
g
th
e
v
er
i
f
i
c
ati
o
n
e
qu
at
i
o
n
is
*
m
o
d
.
e
SE
R
y
p
=
4.
A
n
al
y
s
is
o
f
t
h
e
P
r
o
p
o
se
d
Blind
S
ign
atu
r
e S
ec
u
r
it
y
4.1
. Co
r
r
e
ctn
es
s
T
he
ore
m
1:
T
he
s
i
gn
at
ure
(
,
)
ES
i
s
a
v
a
l
i
d
b
l
i
nd
s
i
gn
atu
r
e
s
c
he
m
e
c
orr
es
po
nd
i
ng
to
the
m
es
s
ag
e
M
.
-
B
l
i
nd
s
i
gn
atu
r
e
S
c
he
m
e:
P
r
oo
f
: In
ac
c
ordanc
e
wi
th
t
he
r
ou
nd
s
4,
5 a
nd
6
w
e
ha
v
e
(
)
(
)
m
o
d
.
e
e
d
e
e
e
e
e
D
DD
S
D
n
+
+
Us
i
ng
t
he
c
o
nd
i
ti
o
n
(
)
m
o
d
,
e
S
D
n
+
c
orr
ec
tne
s
s
of
the
s
c
he
m
e
i
s
prov
ed
as
f
ol
l
o
w
s
:
*
*
(
m
o
d
)
.
e
S
E
D
E
D
E
R
y
y
y
y
R
y
p
E
E
+
+
=
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
5,
O
c
tob
er 20
19
:
23
2
7
-
23
34
2332
-
B
l
i
nd
m
ul
ti
-
s
i
gn
a
ture
S
c
he
m
e:
Ins
tea
d
of
the
v
a
l
ue
D
and
R
i
n
the
proof
eq
ua
t
i
on
s
i
n
the
b
l
i
nd
s
i
gn
at
ure
s
c
he
m
e,
i
t
i
s
r
ep
l
ac
ed
b
y
t
he
v
a
l
ue
D
and
R
.
T
he
r
es
ul
t
i
s
l
i
k
e
the
bl
i
nd
s
i
gn
at
ure s
c
he
m
e.
4.2
. Unl
inkabil
it
y
In
a
bl
i
nd
s
i
gn
a
ture
s
c
he
m
e,
the
u
nl
i
nk
ab
i
l
i
t
y
prop
ert
y
(
or
bl
i
nd
ne
s
s
propert
y
)
m
ak
es
i
t
i
m
po
s
s
i
bl
e
f
or
th
e
s
i
gn
er
to
d
eri
v
e
t
he
l
i
nk
be
t
w
e
en
a
gi
v
en
s
i
g
na
t
ure
a
nd
th
e
i
ns
tan
c
e
of
the
s
i
gn
i
ng
s
c
h
em
e
w
h
i
c
h
produc
es
t
he
bl
i
nd
ed
f
orm
of
tha
t
s
i
gn
at
ure.
T
he
orem
2:
T
he
s
c
he
m
e
prov
i
de
s
u
nl
i
nk
ab
i
l
i
t
y
pro
p
ert
y
i
n
th
e
c
as
e
w
h
en
t
he
m
e
s
s
ag
e
M
an
d
s
i
g
na
t
ure
(
,
)
ES
wi
l
l
b
e
pres
en
te
d t
o
the
s
i
gn
er.
-
B
l
i
nd
s
i
gn
atu
r
e
s
c
h
em
e:
P
r
oo
f
:
W
i
th
eq
ua
l
prob
a
bi
l
i
t
y
of
ea
c
h
of
the
us
e
r
s
,
w
h
o
pa
r
ti
c
i
pa
ted
i
n
t
he
bl
i
n
d
s
i
gn
at
ure
s
c
he
m
e,
the
y
c
ou
l
d
pro
v
i
de
a
s
i
gn
at
ure
on
a
do
c
um
en
t
M
.
T
hi
s
c
a
n
l
ea
d
to
the
f
ol
l
o
wi
ng
s
tat
em
en
t:
f
r
om
the
f
ac
t
that
an
y
tr
i
p
l
e
(
,
,
)
R
D
E
f
r
o
m
the
s
et
of
s
uc
h
tr
i
p
l
e
s
f
or
m
ed
b
y
t
he
s
i
g
ne
r
m
a
y
be
as
s
oc
i
ate
d
wi
th
th
e
s
i
g
na
tu
r
e
(
,
)
ES
of
thi
s
do
c
um
en
t
M
.
Ind
e
ed
,
s
i
nc
e
m
od
DE
R
y
p
=
(
s
ee
r
ou
nd
3
of
the
s
c
he
m
e)
an
d
m
o
d
,
e
SE
R
y
p
=
the
n
the
r
e
l
at
i
on
:
(
m
o
d
)
.
e
S
D
E
E
R
y
y
p
R
−
−
S
o,
when
c
h
o
os
i
ng
r
a
nd
om
eq
ui
prob
ab
l
e
v
a
l
ue
s
an
d
,
th
e
s
i
gn
a
ture
(
,
)
ES
wi
th
eq
ua
l
pr
ob
a
bi
l
i
t
y
c
o
ul
d
be
ge
ne
r
at
ed
wi
th
a
n
y
us
er
i
n
the
proc
es
s
of
bl
i
n
d s
i
g
ni
ng
.
-
B
l
i
nd
m
ul
ti
-
s
i
g
na
t
ure
s
c
he
m
e:
Ins
tea
d
of
the
v
a
l
u
e
D
and
R
i
n
the
proof
eq
u
ati
o
ns
i
n
the
bl
i
nd
s
i
gn
atu
r
e
s
c
he
m
e,
i
t
i
s
r
e
pl
ac
ed
b
y
the
v
al
u
e
D
an
d
R
.
T
he
r
es
ul
t
i
s
l
i
k
e
the
b
l
i
nd
s
i
gn
a
ture s
c
he
m
e.
4.3
. R
a
n
d
o
miz
atio
n
T
he
s
i
gn
er
ha
d
be
tte
r
i
nj
ec
t
on
e
or
m
ore
r
an
do
m
i
z
i
ng
f
ac
tors
i
nto
the
bl
i
nd
ed
m
es
s
ag
e
s
uc
h
tha
t
t
he
att
ac
k
ers
c
an
no
t
pred
i
c
t
th
e
ex
ac
t
c
on
te
nt
of
the
m
es
s
ag
e
th
e
s
i
gn
er
s
i
gn
s
.
T
he
ore
m
3:
T
he
s
c
he
m
e p
r
ov
i
de
s
r
a
nd
om
i
z
at
i
o
n p
r
op
ert
y
.
-
B
l
i
nd
s
i
gn
atu
r
e
s
c
h
em
e:
P
r
oo
f
:
In
t
he
pr
op
os
e
d
s
c
he
m
e,
att
ac
k
ers
are
i
nf
ea
s
i
b
l
e
t
o
s
i
g
n
a
v
a
l
i
d
s
i
g
na
tur
e
(
,
)
ES
on
be
ha
l
f
of
the
orig
i
na
l
s
i
gn
e
r
.
T
he
s
i
gn
er
s
el
ec
ts
a
r
an
do
m
v
al
ue
11
kn
−
an
d
c
om
pu
tes
m
od
k
Rp
=
an
d
s
en
ds
R
to
th
e
us
er
A
.
T
o
g
et
a
r
an
do
m
v
a
l
ue
k
f
r
o
m
R
i
s
c
o
m
pu
tat
i
on
a
l
l
y
i
nf
ea
s
i
b
l
e
(
i
t
i
s
di
f
f
i
c
ul
t
to
d
ete
r
m
i
ne
k
b
ec
au
s
e
tha
t
the
d
eri
v
at
i
on
i
s
s
ol
v
i
ng
the
di
s
c
r
ete
l
o
ga
r
i
thm
probl
em
)
.
T
he
r
ef
ore,
i
n
th
e
prop
os
ed
s
c
h
em
e,
att
ac
k
er
s
c
an
no
t
r
em
ov
e t
he
r
a
nd
om
k
f
r
o
m
th
e c
orr
es
po
nd
i
n
g s
i
g
na
t
ure
(
,
)
ES
of
m
es
s
ag
e
M
.
-
B
l
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e:
Ins
tea
d
of
the
v
al
ue
k
an
d
R
i
n
th
e
pr
oo
f
eq
ua
t
i
on
s
i
n
the
b
l
i
nd
s
i
gn
at
ure
s
c
he
m
e,
i
t
i
s
r
ep
l
ac
ed
b
y
t
he
v
a
l
u
e
i
k
and
i
R
.
T
he
r
es
ul
t
i
s
l
i
k
e
th
e
bl
i
n
d
s
i
gn
at
ure s
c
he
m
e.
4.4.
Un
f
o
r
g
ea
b
ilit
y
It
m
ea
ns
tha
t
on
l
y
t
he
s
i
gn
er
c
an
ge
ne
r
ate
the
v
a
l
i
d
s
i
gn
atu
r
e.
T
he
i
ntrud
er
m
a
y
att
ac
k
the
propos
e
d
s
c
he
m
e
b
y
f
o
l
l
o
wi
n
g
wa
y
.
In
tr
ud
er
tr
i
es
t
o
de
r
i
v
e
t
he
s
i
gn
atu
r
e
(
,
)
ES
f
or
a
g
i
v
en
m
e
s
s
ag
e
M
b
y
l
ett
i
ng
on
e
i
nt
eg
er
f
i
x
ed
an
d
f
i
nd
i
n
g
the
oth
er
on
e.
F
or
ex
a
m
pl
e,
i
ntrud
e
r
s
el
ec
ts
E
an
d
tr
i
es
t
o
f
i
g
ure
ou
t
t
he
v
a
l
u
e
of
S
s
ati
s
f
y
i
n
g
m
o
d
e
SE
R
y
p
=
an
d
v
i
s
e
-
v
ers
a.
T
o
do
thi
s
,
i
ntrud
er
f
i
r
s
t
c
ho
os
es
at
r
an
do
m
an
i
nte
g
er
R
.
He
the
n
c
om
pu
tes
l
o
g
m
o
d
eE
S
R
y
p
−
=
and
on
l
y
i
f
two
d
i
f
f
i
c
ul
t p
r
ob
l
em
s
i
s
break
ab
l
e.
4.5
.
P
e
r
f
o
r
man
c
e
T
he
s
ec
urit
y
of
the
n
e
w
bl
i
n
d
di
g
i
ta
l
s
i
gn
atu
r
e
s
c
he
m
e h
as
be
en
pr
ov
en
to
be
eq
u
i
v
a
l
en
t
to
s
ol
v
i
ng
t
wo
i
nd
ep
e
nd
en
t
d
i
f
f
i
c
ul
t
prob
l
em
s
s
i
m
ul
tan
eo
us
l
y
i
nc
l
ud
i
ng
I
F
P
an
d
DL
P
.
W
e
i
nv
es
ti
ga
te
the
pe
r
f
orm
an
c
e
of
ou
r
s
c
he
m
es
i
n
the
nu
m
be
r
of
m
od
ul
ar
m
ul
ti
p
l
i
c
a
ti
o
n,
nu
m
be
r
of
ha
s
hi
n
g
o
pe
r
a
ti
on
,
n
um
be
r
of
r
an
do
m
nu
m
be
r
ge
ne
r
a
ti
o
n,
nu
m
be
r
of
i
n
v
ers
e
c
o
m
pu
tat
i
on
s
, n
um
be
r
of
c
ub
e ro
ot
a
nd
nu
m
be
r
of
m
o
du
l
ar ex
po
ne
nt
i
at
i
o
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
B
l
i
nd
mu
l
ti
-
s
i
gn
a
ture s
c
he
me
b
as
ed
o
n
f
ac
tori
n
g
....
(
Duc
Ngu
y
e
n Tan)
2333
T
i
m
e
f
or
c
o
m
pu
ti
ng
m
od
ul
ar
ad
d
i
ti
on
an
d
s
u
btrac
ti
o
n
are
i
gn
ored,
s
i
nc
e
i
t
i
s
m
u
c
h
s
m
al
l
er
t
ha
n
t
i
m
e
f
or
c
o
m
pu
ti
ng
m
od
ul
ar
ex
po
n
en
t
i
a
ti
on
,
m
od
ul
ar
m
ul
ti
p
l
i
c
at
i
o
n
an
d
m
od
ul
ar
i
n
v
ers
e.
T
he
c
om
pa
r
i
s
on
s
of
c
om
pu
tat
i
o
n
c
os
ts
pe
r
f
or
m
ed
b
y
the
us
er,
s
i
g
n
er
an
d
v
er
i
f
i
er
be
t
w
e
en
t
he
prop
os
ed
bl
i
nd
s
i
g
na
t
ure
s
c
he
m
e
an
d
the
s
c
he
m
e
of
[27
]
are
s
u
m
m
ariz
ed
i
n
T
ab
l
e 1
a
nd
T
ab
l
e 2
.
T
ab
l
e 1
. T
he
C
om
pu
tat
i
o
n
Cos
ts
of
th
e
P
r
op
os
e
d B
l
i
n
d M
u
l
ti
-
si
gn
a
ture
S
c
he
m
e
an
d t
he
S
c
he
m
e
of
[2
7]
Ty
p
e
o
f
Op
e
r
a
t
ion
s
P
e
r
f
o
r
m
e
d
b
y
t
h
e
u
s
e
r
P
e
r
f
o
r
m
e
d
b
y
t
h
e
s
ign
e
r
Ou
r
s
c
h
e
m
e
[
2
7
]
Ou
r
s
c
h
e
m
e
[
2
7
]
N
u
m
b
e
r
s
o
f
E
x
p
o
n
e
n
t
iat
ion
s
N
u
m
b
e
r
s
o
f
I
n
v
e
r
s
e
s
N
u
m
b
e
r
s
o
f
H
a
s
h
ing
s
N
u
m
b
e
r
s
o
f
M
u
lt
ipli
c
a
t
ion
s
N
u
m
b
e
r
s
o
f
c
u
b
e
r
o
o
t
R
a
n
d
o
m
n
u
mbe
r
g
e
n
e
r
a
t
ion
3
1
1
3
0
3
2
1
1
5
0
3
2
0
0
2
0
1
1
0
0
1
1
1
T
ab
l
e 2
. T
he
C
om
pu
tat
i
o
n
Cos
ts
of
th
e
P
r
op
os
e
d B
l
i
n
d M
u
l
ti
-
s
i
gn
a
ture
S
c
he
m
e
an
d t
he
S
c
he
m
e
of
[2
7]
Ty
p
e
o
f
Op
e
r
a
t
ion
s
P
e
r
f
o
r
m
e
d
b
y
t
h
e
v
e
r
if
ier
Ou
r
s
c
h
e
m
e
[
2
7
]
N
u
m
b
e
r
s
o
f
E
x
p
o
n
e
n
t
iat
ion
s
N
u
m
b
e
r
s
o
f
H
a
s
h
ing
s
N
u
m
b
e
r
s
o
f
M
u
lt
ipli
c
a
t
ion
s
3
1
1
2
1
3
T
hi
s
s
ec
ti
on
wi
l
l
c
om
pa
r
e
the
pe
r
f
or
m
an
c
e
of
our
bl
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e
w
i
t
h
the
bl
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e
i
n
[27
]
a
l
s
o
d
es
i
g
n
th
e
b
l
i
nd
m
ul
ti
-
s
i
g
na
t
ure
s
c
he
m
e,
bu
t
the
ba
s
i
c
s
c
he
m
e
i
s
ba
s
e
d
on
th
e
R
ab
i
n
an
d
t
he
S
c
h
no
r
r
s
c
he
m
es
an
d
us
i
n
g
S
3
i
ns
tea
d
S
i
n
the
S
i
g
na
tur
e
v
erif
i
c
ati
on
proc
ed
ure.
O
ur
bl
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e
i
s
ba
s
ed
on
t
he
R
S
A
an
d t
he
S
c
hn
orr
s
c
he
m
es
a
nd
us
i
ng
S
e
i
ns
te
ad
S
i
n t
h
e
S
i
gn
at
ure
v
erif
i
c
at
i
on
proc
ed
ure.
F
r
o
m
the
c
om
pa
r
i
s
on
T
ab
l
e
1
a
nd
T
ab
l
e
2,
we
r
ea
l
i
z
e
d
t
ha
t
t
he
t
i
m
e
c
os
ts
o
f
the
propos
e
d
b
l
i
nd
m
ul
ti
-
s
i
gn
at
ure
s
c
he
m
e
ha
s
m
ore
the
t
i
m
e
c
os
t
tha
n
th
e
s
c
he
m
e
i
n
[27
]
w
i
th
pe
r
f
or
m
ed
b
y
ue
r
a
nd
wi
th
pe
r
f
or
m
ed
b
y
the
V
erif
i
er.
Ho
w
e
v
er,
wi
t
h
p
erf
or
m
ed
b
y
t
he
s
i
gn
er,
i
t
i
s
ea
s
i
er
to
pe
r
f
orm
be
c
au
s
e
i
t
i
s
no
t
r
eq
u
i
r
ed
to
ex
t
r
ac
t
the
s
q
ua
r
e
3
r
o
ot
t
o
c
a
l
c
ul
a
te
t
he
D
v
a
l
ue
(
D
i
s
us
e
d
to
c
om
pu
ti
ng
the
bl
i
n
d
s
i
g
na
t
ure
S
).
A
nd
th
eref
ore,
th
e
y
c
an
be
ap
p
l
i
ed
i
n p
r
ac
t
i
c
e.
5
.
Co
n
clus
ion
In
th
i
s
pa
p
er,
w
e
pr
op
os
e
d
a
ne
w
s
i
gn
atu
r
e
s
c
he
m
e
f
r
o
m
tw
o
d
i
f
f
i
c
ul
t
probl
e
m
s
IFP
an
d
DL
P
.
T
he
n
ex
p
an
di
ng
t
o
prop
os
e
a
s
i
ng
l
e
bl
i
n
d
s
i
gn
atu
r
e
s
c
he
m
e
an
d
a
bl
i
nd
m
ul
ti
-
s
i
gn
atu
r
e
s
c
he
m
e,
whi
c
h
r
e
qu
i
r
es
the
s
i
m
ul
tan
e
ou
s
break
i
ng
o
f
two
i
nd
ep
e
nd
e
nt
di
f
f
i
c
ul
t
prob
l
em
s
,
the
s
e
are
ba
s
ed
on
th
e
RS
A
s
i
gn
at
ure
s
c
he
m
e
an
d
S
c
hn
orr
s
i
gn
at
ure
s
c
he
m
e.
It
ha
s
be
e
n
pro
v
ed
to
be
c
orr
ec
t,
bl
i
n
d,
u
nf
orged,
r
an
d
om
an
d
pro
v
i
d
es
hi
g
he
r
l
e
v
el
s
ec
urit
y
t
ha
n
s
c
he
m
es
tha
t
ba
s
ed
on
a
s
i
n
gl
e
ha
r
d
pro
bl
em
.
T
he
r
es
ul
ts
s
ho
w
t
ha
t
the
pr
op
os
e
d
bl
i
nd
m
ul
ti
-
s
i
g
na
t
ure
s
i
g
na
t
ure
s
c
he
m
e
are
s
af
e
an
d
pres
en
t
h
i
g
h
pe
r
f
orm
an
c
e;
the
r
ef
ore,
th
e
y
c
an
be
a
pp
l
i
e
d
i
n
prac
ti
c
e
s
uc
h
as
the
prop
os
ed
s
c
h
em
es
c
an
be
ap
pl
i
ed
i
n
e
l
ec
ti
on
s
y
s
t
em
s
an
d
di
g
i
ta
l
c
as
h s
c
he
m
es
.
Ref
er
en
ce
s
[1
]
M
e
n
e
z
e
s
A
J
,
Va
n
s
to
n
e
SA.
H
a
n
d
b
o
o
k
o
f
Ap
p
l
i
e
d
Cry
p
to
g
r
a
p
h
y
.
CR
C Pre
s
s
.
1
9
9
6
:
7
8
0
.
[2
]
Z
Sh
a
o
.
S
e
c
u
r
i
ty
o
f
a
n
e
w
d
i
g
i
ta
l
s
i
g
n
a
tu
re
s
c
h
e
m
e
b
a
s
e
d
o
n
fa
c
to
r
i
n
g
a
n
d
d
i
s
c
re
t
e
l
o
g
a
ri
t
h
m
s
.
In
te
rn
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
Com
p
u
te
r M
a
th
e
m
a
ti
c
s
.
2
0
0
5
;
8
2
(1
0
)
:
1215
-
1
2
1
9
.
[3
]
T
H
Che
n
,
W
B
L
e
e
,
G
Horn
g
.
Rem
a
rk
s
o
n
s
o
m
e
s
i
g
n
a
tu
r
e
s
c
h
e
m
e
s
b
a
s
e
d
o
n
fa
c
to
ri
n
g
a
n
d
d
i
s
c
re
te
l
o
g
a
r
i
th
m
s
.
A
p
p
l
i
e
d
M
a
th
e
m
a
ti
c
s
a
n
d
Co
m
p
u
ta
ti
o
n
.
2005
:
1
0
7
0
-
1
0
7
5
.
[4
]
J
Bu
c
h
m
a
n
n
,
A
M
a
y
,
U
Vo
l
l
m
e
r.
Pe
rs
p
e
c
t
i
v
e
s
fo
r
c
ry
p
to
g
r
a
p
h
i
c
l
o
n
g
te
rm
s
e
c
u
ri
t
y.
Com
m
u
n
i
c
a
t
i
o
n
s
o
f
th
e
ACM
.
2
0
0
6
;
49(
9)
:
50
-
5
5
.
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
5,
O
c
tob
er 20
19
:
23
2
7
-
23
34
2334
[5
]
D
Cha
u
m
.
Bl
i
n
d
s
i
g
n
a
tu
re
s
f
o
r
u
n
tr
a
c
e
a
b
l
e
p
a
y
m
e
n
t
s
.
Ad
v
a
n
c
e
s
i
n
Cry
p
to
l
o
g
y
,
CR
Y
PT
O
’
8
2
.
1
9
8
2
:
199
-
2
0
3
.
[6
]
N
M
F
T
a
h
a
t,
S
M
A
Sh
a
tn
a
w
i
,
ES
Is
m
a
i
l
.
New
Pa
rti
a
l
l
y
Bl
i
n
d
Si
g
n
a
t
u
re
Ba
s
e
d
o
n
Fa
c
to
ri
n
g
a
n
d
Dis
c
re
te
L
o
g
a
r
i
th
m
s
.
J
o
u
r
n
a
l
o
f
M
a
th
e
m
a
ti
c
s
a
n
d
St
a
ti
s
ti
c
s
.
2
0
0
8
;
4
(
2
):
1
2
4
-
1
2
9
.
[7
]
N
M
F
T
a
h
a
t,
ES
Is
m
a
i
l
,
RR
Ah
m
a
d
.
A
N
e
w
Bl
i
n
d
Si
g
n
a
tu
r
e
S
c
h
e
m
e
B
a
s
e
d
O
n
Fa
c
to
r
i
n
g
a
n
d
Di
s
c
re
t
e
L
o
g
a
ri
th
m
s
.
I
n
te
rn
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
Cry
p
t
o
l
o
g
y
Re
s
e
a
rc
h
.
2
0
0
9
;
1
(
1
):
1
-
9.
[8
]
HF
Hua
n
g
,
CC.
Cha
n
g
.
A
n
e
w
d
e
s
i
g
n
o
f
e
ff
i
c
i
e
n
t
b
l
i
n
d
s
i
g
n
a
tu
re
s
c
h
e
m
e
.
Th
e
J
o
u
rn
a
l
o
f
Sy
s
te
m
s
a
n
d
So
f
twa
re
.
2
0
0
4
:
73
:
3
9
7
-
4
0
3
.
[9
]
J
L
Cam
e
n
i
s
h
,
JM
Pri
v
e
te
a
u
,
MA
Sta
d
l
e
r
.
Bl
i
n
d
s
i
g
n
a
tu
re
b
a
s
e
d
o
n
th
e
d
i
s
c
re
t
e
l
o
g
a
ri
th
m
p
ro
b
l
e
m
.
Ad
v
a
n
c
e
s
i
n
Cry
p
to
l
o
g
y
(Eu
ro
c
ry
p
t
'
9
4
),
L
NC
S 9
5
0
,
S
p
r
i
n
g
e
r
-
Ve
rl
a
g
.
1
9
9
4
:
4
2
8
-
4
3
2
.
[1
0
]
D
J
e
n
a
,
SK
J
e
n
a
,
B
M
a
j
h
i
,
SK
Pa
n
i
g
ra
h
y
.
A
n
o
v
e
l
ECD
L
P
-
b
a
s
e
d
b
l
i
n
d
s
i
g
n
a
tu
re
s
c
h
e
m
e
w
i
th
a
n
i
l
l
u
s
tra
t
i
o
n
.
W
e
b
e
n
g
i
n
e
e
ri
n
g
a
n
d
a
p
p
l
i
c
a
t
i
o
n
s
.
2
0
0
8
:
59
-
6
8
.
[1
1
]
D
Zh
e
n
g
,
K
Ch
e
n
,
W
Q
i
u
.
Ne
w
Rab
i
n
-
l
i
k
e
s
i
g
n
a
tu
r
e
s
c
h
e
m
e
.
W
o
rk
s
h
o
p
Pro
c
e
e
d
i
n
g
s
o
f
t
h
e
Se
v
e
n
t
h
In
te
rn
a
ti
o
n
a
l
Con
fe
re
n
c
e
o
n
Dis
tri
b
u
te
d
M
u
l
ti
m
e
d
i
a
Sy
s
te
m
s
,
Kn
o
w
l
e
d
g
e
Sy
s
te
m
s
I
n
s
t
i
tu
te
.
2001
:
185
-
1
8
8
.
[1
2
]
F
G
J
e
n
g
,
T
L
Che
n
,
T
S
Che
n
.
An
EC
C
-
Ba
s
e
d
Bl
i
n
d
Si
g
n
a
t
u
re
Sc
h
e
m
e
.
J
o
u
r
n
a
l
o
f
n
e
tw
o
rk
s
.
2
0
1
0
;
5
(
8
)
:
9
2
1
-
9
2
8
.
[1
3
]
C
Po
p
e
s
c
u
.
Bl
i
n
d
S
i
g
n
a
tu
re
a
n
d
BM
S
Us
i
n
g
E
l
l
i
p
ti
c
Curv
e
s
.
Stu
d
i
a
u
n
i
v
.
“b
a
b
e
s
–
b
o
l
y
a
i
”,
I
n
fo
rm
a
ti
c
a
.
1999
:
43
-
4
9
.
[1
4
]
SS
Cho
w
,
e
t
a
l
.
Tw
o
Im
p
ro
v
e
d
Pa
rti
a
l
l
y
Bl
i
n
d
Si
g
n
a
tu
r
e
Sc
h
e
m
e
s
fro
m
Bi
l
i
n
e
a
r
Pa
i
r
i
n
g
s
.
In
fo
rm
a
ti
o
n
Se
c
u
ri
ty
a
n
d
Pr
i
v
a
c
y
.
2
0
0
5
;
3
5
4
7
:
3
1
6
-
3
2
8
.
[1
5
]
NA
M
o
l
d
o
v
y
an
.
Bl
i
n
d
Si
g
n
a
t
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r
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Pro
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o
l
s
fr
o
m
Di
g
i
ta
l
Si
g
n
a
tu
re
Sta
n
d
a
r
d
s
.
In
te
r
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Net
work
Se
c
u
ri
ty
.
2
0
1
1
:
1
3
(1
):
22
-
30
.
[1
6
]
T
A
T
Ngu
y
e
n
,
T
K
Dan
g
.
En
h
a
n
c
e
d
s
e
c
u
ri
ty
i
n
i
n
t
e
rn
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v
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n
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p
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to
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l
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n
d
s
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g
n
a
tu
re
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n
d
d
y
n
a
m
i
c
b
a
l
l
o
ts
.
E
l
e
c
tro
n
i
c
Co
m
m
e
rc
e
Re
s
e
a
r
c
h
.
2
0
1
3
;
13
(3
):
2
5
7
-
272
.
[1
7
]
S
Ve
rm
a
,
B
K
Sh
a
r
m
a
l
.
Ne
w
Pro
x
y
Bl
i
n
d
M
u
l
ti
Si
g
n
a
tu
re
b
a
s
e
d
o
n
I
n
te
g
e
r
Fa
c
to
ri
z
a
ti
o
n
a
n
d
Dis
c
re
te
-
L
o
g
a
ri
th
m
Pro
b
l
e
m
s
.
2012
;
1
(3
)
:
185
-
190.
[1
8
]
S
Pa
n
d
a
,
e
t
a
l
.
An
Ap
p
l
i
c
a
ti
o
n
o
f
ti
m
e
s
ta
m
p
e
d
p
ro
x
y
b
l
i
n
d
s
i
g
n
a
t
u
r
e
i
n
e
-
v
o
ti
n
g
.
In
te
rn
a
ti
o
n
a
l
J
o
u
rn
a
l
o
n
Co
m
p
u
te
r
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i
e
n
c
e
a
n
d
En
g
i
n
e
e
r
i
n
g
.
2
0
1
3
;
5
(6
)
:
5
4
7
-
5
5
2
.
[1
9
]
H Su
n
,
Y
Ge
.
New
Ce
rti
fi
c
a
t
e
l
e
s
s
B
l
i
n
d
Ri
n
g
Si
g
n
a
tu
r
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Sc
h
e
m
e
.
TE
L
KO
M
NIKA I
n
d
o
n
e
s
i
a
n
J
o
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r
n
a
l
o
f
El
e
c
tr
i
c
a
l
En
g
i
n
e
e
ri
n
g
.
2
0
1
4
;
12
(
1
):
7
7
8
-
7
8
3
.
[2
0
]
NF
Sh
i
l
b
a
y
e
h
,
RA
Al
-
Sa
i
d
i
,
AH
Al
s
s
w
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y
.
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v
a
l
u
a
ti
o
n
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d
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a
l
y
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s
o
f
th
e
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c
u
r
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E
-
Vo
ti
n
g
Au
th
e
n
t
i
c
a
ti
o
n
Pr
e
p
a
ra
ti
o
n
S
c
h
e
m
e
.
I
n
te
rn
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Com
p
u
te
r
a
n
d
In
fo
rm
a
t
i
o
n
E
n
g
i
n
e
e
ri
n
g
.
2016
:
1
0
(3
)
:
5
6
0
-
5
6
8
.
[2
1
]
H
M
i
n
h
,
N
Hai
,
N
M
o
l
d
o
v
y
a
n
,
T
G
i
a
n
g
.
New
Bl
i
n
d
S
i
g
n
a
t
u
re
Pro
to
c
o
l
s
Ba
s
e
d
o
n
a
New
Hard
Pro
b
l
e
m
.
Th
e
I
n
te
rn
a
ti
o
n
a
l
Ar
a
b
J
o
u
rn
a
l
o
f
I
n
fo
rm
a
ti
o
n
T
e
c
h
n
o
l
o
g
y
.
2
0
1
7
;
1
4
(3
)
:
3
0
7
-
313
.
[2
2
]
S
J
a
m
e
s
,
T
G
o
w
ri
,
G
VR
Ba
b
u
,
PV
Red
d
y
.
Id
e
n
ti
ty
-
Ba
s
e
d
Bl
i
n
d
Si
g
n
a
tu
re
Sc
h
e
m
e
w
i
th
M
e
s
s
a
g
e
Rec
o
v
e
ry
,
In
t
e
rn
a
t
i
o
n
a
l
J
o
u
r
n
a
l
o
f
E
l
e
c
tri
c
a
l
a
n
d
Com
p
u
te
r
En
g
i
n
e
e
ri
n
g
(
IJ
ECE)
.
2
0
1
7
;
7
(
5
):
2674
-
2
6
8
2
.
[2
3
]
R
Riv
e
s
t,
A
Sh
a
m
i
r,
L
Ad
l
e
m
a
n
.
A
m
e
th
o
d
fo
r
o
b
t
a
i
n
i
n
g
d
i
g
i
ta
l
s
i
g
n
a
tu
re
s
a
n
d
p
u
b
l
i
c
k
e
y
c
ry
p
to
s
y
s
t
e
m
s
.
Com
m
u
n
i
c
a
ti
o
n
s
o
f
t
h
e
ACM
.
1978;
21
(
2
):
1
2
0
-
1
2
6
.
[2
4
]
Sc
h
n
o
rr
CP
.
Eff
i
c
i
e
n
t
s
i
g
n
a
tu
r
e
g
e
n
e
ra
ti
o
n
b
y
s
m
a
r
t
c
a
rd
s
.
J
o
u
rn
a
l
o
f
Cr
y
p
to
l
o
g
y
.
1
9
9
1
;
4
:
161
-
1
7
4
.
[2
5
]
M
Bl
u
m
.
CR
Y
P
T
O
.
1981
:
11
-
1
5
.
[2
6
]
M
e
n
e
z
e
s
A
J
,
Va
n
s
to
n
e
S
A
.
H
a
n
d
b
o
o
k
o
f
Ap
p
l
i
e
d
Cry
p
to
g
r
a
p
h
y
.
CR
C Pre
s
s
.
1
9
9
6
.
[2
7
]
DN
T
a
n
,
H
N
Na
m
,
M
N
Hie
u
.
Bl
i
n
d
s
i
g
n
a
tu
re
s
c
h
e
m
e
a
n
d
b
l
i
n
d
m
u
l
ti
-
s
i
g
n
a
tu
r
e
s
c
h
e
m
e
b
a
s
e
d
o
n
tw
o
h
a
rd
p
r
o
b
l
e
m
s
.
T
h
e
2
0
t
h
Nat
i
o
n
a
l
Con
f
e
re
n
c
e
o
n
El
e
c
tro
n
i
c
s
,
Co
m
m
u
n
i
c
a
ti
o
n
s
a
n
d
I
n
fo
rm
a
ti
o
n
T
e
c
h
n
o
l
o
g
y
-
REV
-
ECIT
.
2017
:
95
-
1
0
0
.
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